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Properties

Label 2-2940-7.2-c1-0-15
Degree $2$
Conductor $2940$
Sign $0.991 + 0.126i$
Analytic cond. $23.4760$
Root an. cond. $4.84520$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)3-s + (0.5 − 0.866i)5-s + (−0.499 + 0.866i)9-s + (0.822 + 1.42i)11-s + 2.64·13-s + 0.999·15-s + (−0.822 − 1.42i)17-s + (4.14 − 7.18i)19-s + (−0.822 + 1.42i)23-s + (−0.499 − 0.866i)25-s − 0.999·27-s + 7.64·29-s + (−2.14 − 3.71i)31-s + (−0.822 + 1.42i)33-s + (−0.322 + 0.559i)37-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + (0.223 − 0.387i)5-s + (−0.166 + 0.288i)9-s + (0.248 + 0.429i)11-s + 0.733·13-s + 0.258·15-s + (−0.199 − 0.345i)17-s + (0.951 − 1.64i)19-s + (−0.171 + 0.297i)23-s + (−0.0999 − 0.173i)25-s − 0.192·27-s + 1.41·29-s + (−0.385 − 0.667i)31-s + (−0.143 + 0.248i)33-s + (−0.0530 + 0.0919i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2940 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2940\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7^{2}\)
Sign: $0.991 + 0.126i$
Analytic conductor: \(23.4760\)
Root analytic conductor: \(4.84520\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2940} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2940,\ (\ :1/2),\ 0.991 + 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.260592120\)
\(L(\frac12)\) \(\approx\) \(2.260592120\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 \)
good11 \( 1 + (-0.822 - 1.42i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 2.64T + 13T^{2} \)
17 \( 1 + (0.822 + 1.42i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.14 + 7.18i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.822 - 1.42i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 7.64T + 29T^{2} \)
31 \( 1 + (2.14 + 3.71i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.322 - 0.559i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 4.93T + 41T^{2} \)
43 \( 1 + 5.93T + 43T^{2} \)
47 \( 1 + (-3 + 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.64 + 2.85i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.46 - 9.47i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4 + 6.92i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.322 + 0.559i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 13.6T + 71T^{2} \)
73 \( 1 + (-6.61 - 11.4i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (1.14 - 1.98i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 10.9T + 83T^{2} \)
89 \( 1 + (7.11 - 12.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 8T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.718045124419368021657310720833, −8.245290485996162310544261898093, −7.14484594699151467737425480989, −6.58027475734136304564912429914, −5.43682007160980287065159545501, −4.89333694703011103219181705200, −4.03006139015354495812346538081, −3.11748252177924021803271974953, −2.14146467057521404832607223079, −0.835232358830141746612723653538, 1.08502021258373310026843029183, 2.01039179605024059468063670494, 3.22277030184530016251691626455, 3.71832979872469511553207423671, 4.97914168325719814637447816266, 5.96712907808048652741072610139, 6.41946265960291264155252753160, 7.25033908811073648706247908903, 8.167971796394983746754689335065, 8.522166414931828255508611838455

Graph of the $Z$-function along the critical line